James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

96 Chapter 2 Limits and Derivatives
Therefore we have
lim f f sxd 1 5tsxdg − lim f sxd 1 lim f5tsxdg (by Limit Law 1)
x l 22 x l 22 x l 22
− lim f sxd 1 5 lim tsxd (by Limit Law 3)
x l 22 x l 22
− 1 1 5s21d − 24
(b) We see that lim x l 1 f sxd − 2. But lim x l 1 tsxd does not exist because the left and
right limits are different:
lim tsxd − 22 lim
x l 12 tsxd − 21
x l 11 So we can’t use Law 4 for the desired limit. But we can use Law 4 for the one-sided
limits:
lim
x l 1
lim
x l 1
f f sxdtsxdg − lim f sxd ? lim tsxd − 2 ? s22d − 24
2 2 2
x l1
x l1
f f sxdtsxdg − lim f sxd ? lim tsxd − 2 ? s21d − 22
1 1 1
x l1
The left and right limits aren’t equal, so lim x l 1 f f sxdtsxdg does not exist.
(c) The graphs show that
lim
x l 2
x l1
f sxd < 1.4 and lim
x l 2 tsxd − 0
Because the limit of the denominator is 0, we can’t use Law 5. The given limit does not
exist because the denominator approaches 0 while the numerator approaches a nonzero
number.
■
If we use the Product Law repeatedly with tsxd − f sxd, we obtain the following law.
Power Law
6. lim
x la f f sxdgn −
f lim
x la f sxd g n
where n is a positive integer
In applying these six limit laws, we need to use two special limits:
7. lim
x l a
c − c 8. lim
x l a
x − a
These limits are obvious from an intuitive point of view (state them in words or draw
graphs of y − c and y − x), but proofs based on the precise definition are requested in
the exercises for Section 2.4.
If we now put f sxd − x in Law 6 and use Law 8, we get another useful special limit.
9. lim
x l a
x n − a n
where n is a positive integer
A similar limit holds for roots as follows. (For square roots the proof is outlined in
Exercise 2.4.37.)
10. lim s n x − s n a where n is a positive integer
x l a
(If n is even, we assume that a . 0.)
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.